Sound Waves - Science Tutor

Speed of sound

The mechanical vibrations that can be interpreted as sound are able to travel through all forms of matter: gases, liquids, solids, and plasmas. The matter that supports the sound is called the medium. Sound cannot travel through a vacuum.
Speed:
v=√

B

ρ

B - coefficient of stiffness, the bulk modulus (or the modulus of bulk elasticity for gases),
ρ - the density

---

Particle displacement

s=s+mcos(kx−ωt)
sm - amplitude of displacement
k - the wavenumber
x - the distance the point has traveled from the wave\'s source
ω - the angular frequency of the wave
t - the time elapsed

---

Pressure of the medium

Sound is a mechanical wave that is an oscillation of pressure Δp transmitted through a solid, liquid, or gas
Δp=Δpmsin(kx−ωt)
Δpm - pressure amplitude.
Δpm=vρωsm
sm - particle displacement amplitude
v - speed of sound in medium
ρ - density of the medium
k - wavenumber
x - the distance the point has traveled from the wave\'s source
ω - the angular frequency of the wave
t - the time elapsed

---

Interferency

Two traveling waves which exist in the same medium will interfere with each other. If their amplitudes add, the interference is said to be constructive interference, and destructive interference if they are \"out of phase\" and subtract.
Φ=

ΔL

Λ

2Π
ΔL - the difference in distance defeated by the waves to reach a common point
Λ - waves length

When the phase difference of the two waves is an integral multiple of 2Π, theinterference is constructive:
Φ=m2Π
m - integral number

Fully destructive interference (both wave expires) occurs when the phase difference is equal to an odd multiple ofΠ:
Φ=(2m+1)Π
m - integral number

---

Sound intensity

The intensity is the product of the sound pressure and the particle velocity:
I=pv, where:
p - sound pressure
v - speed of sound
It is also defined as:
I=

P

S

, where
P - acoustic power
S - sound receiving area

The intensity at a distance r from a point source that emits a sound wave is equal:
I=

P

4Πr2

P - acoustic power

The intensity and amplitude of sound wave displacement sm involves the dependence:
I=

1

2

ρvω2sm2
ρ - density of medium
v - speed of sound
ω - the angular frequency of the wave

---

Sound intensity level

L=10log

I

I0

,
I0=10−12

W

m2

 - standard reference sound intensity,
I - sound intensity

---

Doppler Effect

The perceived frequency f´ is related to the actual frequency f0 and the relative speeds of the source vs, observer vo, and the speed v of waves in the medium by
f′=f0

v(+/−)v0

v(+/−)v0

The choice of using the plus (+) or minus (-) sign is made according to the convention that if the source and observer are moving towards each other the perceived frequency f´ is higher than the actual frequency f0. Likewise, if the source and observer are moving away from each other the perceived frequency f´ is lower than the actual frequency f0.

---